Water jet cutter

Homework Help
1

Water jets are being used more and more for metal cutting operations. If a pump generates a flow of 1 gpm through an orifice of 0.01 in. diameter, what is the average jet speed? What force (lbf) will the jet produce at impact, assuming as an approximation that the water sprays sideways after impact?

2

PK Bot’s answer:

  1. ENGINEERING CONTEXT: This problem addresses waterjet cutting operations in which a high-pressure water jet is forced through a small orifice to generate a high-speed jet capable of impacting material for cutting.

  2. PHYSICS: The dominant physics is conservation of mass and momentum in an inviscid free jet, where the jet speed is set by the volumetric flow rate and the orifice cross-sectional area, and the impact force is determined by the momentum flux of the jet.

  3. ASSUMPTIONS: We assume steady, incompressible, inviscid flow and that upon impact the water “sprays sideways,” so that the entire momentum flux is redirected to impact force; these assumptions are reasonable for an initial engineering estimate in waterjet cutting applications.

  4. SOLUTION TECHNIQUE:

(a) Convert flow rate and orifice diameter to SI units.

• Flow rate: Q=1~\text{gpm}\;=\;\frac{3.785\times10^{-3}~m^3}{60~s}\;\approx\;6.31\times10^{-5}~m^3/s.

• Diameter: d=0.01~in\;=\;0.01\times0.0254~m\;=\;2.54\times10^{-4}~m,
so area: A=\frac{\pi d^2}{4}\;=\;\frac{\pi (2.54\times10^{-4})^2}{4}\;\approx\;5.06\times10^{-8}~m^2.

(b) Average jet speed from conservation of mass:

U=\frac{Q}{A}=\frac{6.31\times10^{-5}}{5.06\times10^{-8}}\approx1246~m/s\,.

(c) The impact force is estimated as the momentum flux:

F=\rho\,Q\,U\quad\text{with}\quad \rho=1000~kg/m^3\,,

thus

F=1000\times6.31\times10^{-5}\times1246\approx78.5~N\,.

(d) Converting the force to pounds-force:

F_{lbf}=\frac{78.5}{4.448}\approx17.6~lbf\,.
  1. REFLECTION: The analysis shows that an extremely small orifice when supplied with a moderate flow rate can generate very high jet speeds (about 1246~m/s), resulting in an impact force of roughly 17.6~lbf; although the analysis neglects viscous and compressibility effects near these extreme speeds, it provides a first-order engineering estimate for the performance of a waterjet used in metal cutting.

To formulate this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 5 How Stuff Works: How waterjet cutters work (HIGHLY RELEVANT)
  2. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 2 (HIGHLY RELEVANT)
  3. Chapter 5 Fundamentals: Forming a free jet (HIGHLY RELEVANT)
  4. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 3 Part 1 (HIGHLY RELEVANT)
  5. Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 4 Part 1 (HIGHLY RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.